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Distributions of cherries for two models of trees. (English) Zbl 0947.92021

Summary: Null models for generating binary phylogenetic trees are useful for testing evolutionary hypotheses and reconstructing phylogenies. We consider two such null models – the Yule and uniform models – and in particular the induced distribution they generate on the number \(C_n\) of cherries in the tree, where a cherry is a pair of leaves each of which is adjacent to a common ancestor. By realizing the process of cherry formation in these two models by extended Polya urn models we show that \(C_n\) is asymptotically normal. We also give exact formulas for the mean and standard deviation of the \(C_n\) in these two models. This allows simple statistical tests for the Yule and uniform null hypotheses.

MSC:

92D15 Problems related to evolution
62P10 Applications of statistics to biology and medical sciences; meta analysis
05C90 Applications of graph theory
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[1] Kubo, T.; Iwasa, Y., Inferring the rates of branching and extinction from molecular phylogenies, Evolution, 49, 694 (1995)
[2] Raup, D. M.; Gould, S. J.; Schopf, T. J.M.; Simberloff, D. S., Stochastic models of phylogeny and the evolution of diversity, J. Geol., 81, 5, 525 (1973)
[3] Gould, S. J.; Raup, D. M.; Sepkowski, J. J.; Schopf, T. J.M.; Simberloff, D. S., The shape of evolution: a comparison of real and random clades, Paleobiology, 3, 23 (1977)
[4] Savage, H. M., The shape of evolution: systematic tree topology, Biolog. J. Linnean Soc., 20, 225 (1983)
[5] Slowinski, J. B.; Guyer, C., Testing the stochasticity of patterns of organismal diversity: an improved null model, Am. Natural., 134, 6, 907 (1989)
[6] Guyer, C.; Slowinski, J. B., Comparisons of observed phylogenetic topologies with null expectations among three monophyletic lineages, Evolution, 45, 2, 340 (1991)
[7] Heard, S. B., Patterns in tree balance among cladistic phenetic and randomly generated phylogenetic trees, Evolution, 46, 6, 1818 (1992)
[8] Kirkpatrick, M.; Slatkin, M., Searching for evolutionary patterns in the shape of a phylogenetic tree, Evolution, 47, 4, 1171 (1993)
[9] Rogers, J. S., Central moments and probability distribution of Colless’s coefficient of tree imbalance, Evolution, 48, 6, 2026 (1994)
[10] Mooers, A., Tree balance and tree completeness, Evolution, 49, 2, 379 (1995)
[11] Mooers, A.; Heard, S. B., Inferring evolutionary process from phylogenetic tree shape, Quart. Rev. Biol., 72, 1, 31 (1997)
[12] R.D.M. Page, E.C. Holmes, Molecular Evolution: a Phylogenetic Approach, ch. 2, Blackwell Science, 1998, pp. 11-36; R.D.M. Page, E.C. Holmes, Molecular Evolution: a Phylogenetic Approach, ch. 2, Blackwell Science, 1998, pp. 11-36
[13] Athreya, K. B.; Karlin, S., Embedding of urn schemes into continuous time markov branching processes and related limit theorems, Ann. Math. Stat., 39, 1801 (1968) · Zbl 0185.46103
[14] K.B. Athreya, P.E. Ney, Branching Processes, Springer, Berlin, 1972, pp. 219-224; K.B. Athreya, P.E. Ney, Branching Processes, Springer, Berlin, 1972, pp. 219-224 · Zbl 0259.60002
[15] Bagchi, A.; Pal, A. K., Asymptotic normality in the generalized Polya-Eggenburger urn model with an application to computer data structures, SIAM J. Algebr. Discr. Meth., 6, 394 (1985) · Zbl 0568.60010
[16] Smythe, R. T., Central limit theorems for urn models, Stochast. Proc. Appl., 65, 115 (1996) · Zbl 0889.60013
[17] Harding, E. F., The probabilities of rooted tree-shapes generated by random bifurcation, Adv. Appl. Probab., 3, 44 (1971) · Zbl 0241.92012
[18] Losos, J. B.; Adler, F. R., Stumped by trees? a generalized null model for patterns of organismal diversity, Am. Natural., 145, 3, 329 (1995)
[19] Nee, S.; May, R. M.; Harvey, P. H., The reconstructed evolutionary process, Philos. Trans. R. Soc. London B, 344, 305 (1994)
[20] Steel, M.; Penny, D., Distribution of tree comparison metrics some new results, System Biol., 42, 2, 126 (1993)
[21] D. Aldous, The continuum random tree II: an overview, in: M.T. Barlow, N.H. Bingham (Eds.), Stochastic Analysis, Cambridge University, Cambridge, 1991, p. 23; D. Aldous, The continuum random tree II: an overview, in: M.T. Barlow, N.H. Bingham (Eds.), Stochastic Analysis, Cambridge University, Cambridge, 1991, p. 23 · Zbl 0791.60008
[22] Aldous, D., The continuum random tree III, Ann. Probab., 21, 248 (1993) · Zbl 0791.60009
[23] Hendy, M. D.; Penny, D., Branch and bound algorithms to determine minimal evolutionary trees, Math. Biosci., 59, 277 (1982) · Zbl 0488.92004
[24] Steel, M. A., Distribution of the symmetric difference metric on phylogenetic trees, SIAM J. Discr. Math., 1, 4, 541 (1988) · Zbl 0675.05024
[25] Härlin, M., Biogeographic patterns and the evolution of Eureptanic nemerteans, Biolog. J. Linnean Soc., 58, 325 (1996)
[26] Huelsenbeck, J. P.; Kirkpatrick, M., Do phylogenetic methods produce trees with biased shapes?, Evolution, 50, 4, 1418 (1996)
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